We construct a spectral triple for the C$^*$-algebra of continuous functions
on the space of $p$-adic integers by using a rooted tree obtained from
coarse-grained approximation of the space, and the forward derivative on the
tree. Additionally, we verify that our spectral triple satisfies the properties
of a compact spectral metric space, and we show that the metric on the space of
$p$-adic integers induced by the spectral triple is equivalent to the usual
$p$-adic metric

Klimek, Slawomir

McBride, Matt